Antarctic Whale Project: Single Species
Ben Weinstein
2016-10-24 16:02:33
0.1 By Month
0.2 CCAMLR Units
0.3 Distance
0.4 Time
0.5 Velocity
0.6 Angles
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
How did the filter change the extent of tracks?
Look at the observations were defined into tracks.
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])
#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha_mu[state[i,g,t-1]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha_mu[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(10, 10)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
##Behavioral States
gamma[1] ~ dbeta(3,2) ## gamma for state 1
dev ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev
#Intercepts
alpha_mu[1] ~ dbeta(1,1)
alpha_mu[2] ~ dbeta(1,1)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)
#latitidunal argos error
argos_alpha~dunif(0,10)
#correlation in argos error
rho ~ dunif(-1, 1)
}"
,fill=TRUE)sink()
## user system elapsed
## 470.616 6.648 78503.8671.4 Chains
## Type Size PrettySize Rows Columns
## jagM rjags.parallel 3180217112 [1] "3 Gb" 6 NA
## data list 61113336 [1] "58.3 Mb" 9 NA
## argos array 39720952 [1] "37.9 Mb" 33 20
## obs array 39720952 [1] "37.9 Mb" 33 20
## j array 19867968 [1] "18.9 Mb" 33 20
## b SpatialPointsDataFrame 16418080 [1] "15.7 Mb" 46421 47
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## m ggmap 13116240 [1] "12.5 Mb" 1280 1280
## d SpatialPointsDataFrame 12825928 [1] "12.2 Mb" 46421 47
## oxy data.frame 12080104 [1] "11.5 Mb" 46421 47## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1787004 95.5 3886542 207.6 3886542 207.6
## Vcells 445620039 3399.9 1016841010 7757.9 877735132 6696.6
Look at the convergence of phi, just for an example
Overall relationship between phi and state, nice test of convergence.
1.4.1 Compare to priors
1.5 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1] 0.53525611 0.46232512 0.60860130
## 2 alpha_mu alpha_mu[2] 0.14592664 0.10773869 0.19289520
## 3 gamma gamma[1] 0.92465146 0.87442694 0.97070532
## 4 gamma gamma[2] 0.15682682 0.11400094 0.19948345
## 5 theta theta[1] -0.01457927 -0.04076998 0.01156162
## 6 theta theta[2] 0.24227712 0.14759083 0.35587027
2 Behavioral Prediction
Relationship between phi and state
2.1 Spatial Prediction
2.2 By individual
Overlay phi and state
2.3 Compared to CMLRR regions
2.4 Autocorrelation in behavior
2.5 Location of Behavior
3 Overlap with Krill Fishery
4 Time spent in grid cell
4.1 ARS
## Type Size PrettySize Rows Columns
## pc tbl_df 598231464 [1] "570.5 Mb" 11490000 10
## a tbl_df 157774144 [1] "150.5 Mb" 3944000 7
## data list 61113336 [1] "58.3 Mb" 9 NA
## argos array 39720952 [1] "37.9 Mb" 33 20
## obs array 39720952 [1] "37.9 Mb" 33 20
## j array 19867968 [1] "18.9 Mb" 33 20
## b SpatialPointsDataFrame 16418080 [1] "15.7 Mb" 46421 47
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## temp ggmap 13116528 [1] "12.5 Mb" 1280 1280
## d SpatialPointsDataFrame 12825928 [1] "12.2 Mb" 46421 47## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1648930 88.1 5103933 272.6 9968622 532.4
## Vcells 119714036 913.4 333198460 2542.2 1009712613 7703.5